In the context of climate change, the impact of extreme precipitation and its chain effects has intensified in the southeastern coastal region of China, posing a serious threat to the socioeconomic development in the region. This study took tropical cyclones–extreme precipitation–flash floods as an example to carry out a risk assessment of flash floods under climate change in the Yantanxi River Basin, southeastern China. To obtain the flash flood inundation characteristics through hydrologic–hydrodynamic modeling, the study combined representative concentration pathway (RCP) and shared socioeconomic pathway (SSP) scenarios to examine the change of flash flood risk and used the geographical detector to explore the driving factors behind the change. The results show that flash flood risk in the Yantanxi River Basin will significantly increase, and that socioeconomic factors and precipitation are the main driving forces. The study highlights the influence of socioeconomic factors on the change of flash flood disaster risk in small river basins. Our findings also provide a reference for regional planning and construction of flood control facilities in flash flood-prone areas, which may help to reduce the risk of flash floods.